ejde.math.txstate.edu · Electronic Journal of Di erential Equations, Vol. 2016 (2016), No. 233, pp. 1{32. ISSN: 1072-6691. URL: or

Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 233, pp. 1–32.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

DYNAMICAL BIFURCATION IN A SYSTEM OF COUPLEDOSCILLATORS WITH SLOWLY VARYING PARAMETERS

IGOR PARASYUK, BOGDAN REPETA

Abstract. This paper deals with a fast-slow system representing n nonlin-early coupled oscillators with slowly varying parameters. We find conditions

which guarantee that all ω-limit sets near the slow surface of the system are

equilibria and invariant tori of all dimensions not exceeding n, the tori of di-mensions less then n being hyperbolic. We show that a typical trajectory

demonstrates the following transient process: while its slow component is farfrom the stationary points of the slow vector field, the fast component exhibits

damping oscillations; afterwards, the former component enters and stays in

a small neighborhood of some stationary point, and the oscillation amplitudeof the latter begins to increase; eventually the trajectory is attracted by an

n-dimesional invariant torus and a multi-frequency oscillatory regime is estab-

lished.

1. Introduction

The coupled oscillators theory plays a significant role in understanding variouspatterns of collective behavior occurring in physical, chemical, biological and socialsystems (see, e. g., [22, 4] and references therein). The variety of behaviors exhibitedby systems of coupled oscillators (SCO) ranges from synchronization to complexchaotic motions. In many cases, transient processes in SCO eventually turn intoself-excited multi-frequency (quasiperiodic) oscillations on toric attractors. Sucha type of behavior in non-conservative systems was observed as early as in the20s–30s of the XX century and since that time was intensively studied (see, e. g.[9, 10, 13, 17, 19, 30, 31, 39, 43, 45]). In the middle of the XX century, there wasdiscovered a phenomenon of a 2-dimensional torus bifurcation accompanying thestability loss of a limit cycle [27, 32, 41]. Later, studies on bifurcations of invarianttori and quasiperiodic oscillations were conducted by many authors (see, e. g.,[6, 7, 8, 14, 20, 24, 29] and references therein) and the actual toolkit for qualitativeinvestigation of such bifurcations was developed in [10, 15, 19, 31, 42, 43].

The aforementioned results concern static bifurcation theory which deals withsystems dependent on time-constant parameters. Within the framework of thistheory, the birth of a stable k-dimensional invariant torus from an equilibrium of

2010 Mathematics Subject Classification. 34C15, 34C23, 34C46, 37D10.Key words and phrases. Dynamical bifurcations; transient processes; invariant tori;

multi-frequency oscillations; coupled oscillators; fast-slow systems.c©2016 Texas State University.

Submitted May 2, 2016. Published August 25, 2016.

1

2 I. PARASYUK, B. REPETA EJDE-2016/233

the system

x = f(x, u), x ∈ Rd, (x :=dxdt

), (1.1)

dependent on the m-dimensional parameter u can be ensured by the following con-ditions: there exists a sufficiently smooth curve (equilibrium curve) x = ξ(s), u =υ(s), s ∈ (−1, 1), such that f(ξ(s), υ(s)) = 0 for all s ∈ (−1, 1); the eigenvaluesof the Jacobi matrix f ′x(ξ(s), υ(s)) have negative real parts for all s ∈ (−1, 0) andpositive real parts for all s ∈ (0, 1); there exists a sufficiently smooth mappingΞ(·, ·) : Tk × [0, 1) → Rn (here Tk := Rk/2πZk denotes the standard torus), suchthat Ξ(ϕ, υ(0)) = ξ(0) for all ϕ ∈ Tk and rank Ξ′ϕ(ϕ, υ(s)) = k for all ϕ ∈ Tk,s ∈ (0, 1); finally, for any s ∈ (0, 1) the toroidal surface x = Ξ(ϕ, υ(s)) is a localattractor of the flow generated by the system x = f(x, υ(s)). Under such condi-tions, when the parameter u, restricted to the curve u = υ(s), passes through thepoint u = υ(0), we observe the stability loss of equilibrium and the birth of a stableinvariant torus. It should be stressed that here the verb “passes” does not have anyrelation to a parameter motion over time.

On the contrary to the theory of static bifurcations, dynamical bifurcation theorydeals with systems which depend on slowly varying in time parameters (fast-slowsystems). Dynamical bifurcation theory focuses on qualitative behavioral transfor-mations which happen in fast-slow systems due to the slow passage of parametersthrough certain critical points in the parameter space. The origin of this theorycan be found in papers on relaxation oscillations (see the review [1]), although theterm “dynamical bifurcation” appeared later, in the 80s of the XX century. Thepapers [47, 33, 34, 35, 5] gave start to studies of actually dynamic bifurcations infast-slow systems. During the last several decades many important results concern-ing the considered area were obtained [12, 23, 11, 46, 40, 3, 26, 18, 28]. Some of themost peculiar features of fast-slow systems, such as the delayed loss of stability, thesynchronization, the existence of the canard solutions and the blue-sky catastrophecan be of great importance in the real-world applications. Nevertheless, some phe-nomena have not yet been fully understood. In particular, as it was noted in [2],this can be said about the emergence of multi-frequency oscillations as a result ofparameters evolution in fast-slow systems.

The present paper grounds on our previous results [37, 38] and aims to fill thegap above. Here we consider the SCO governed by the n-dimensional second ordersystem

w + Ω20(u)w = 2εΛ(u)w + F (w, w, u, µ), (1.2)

dependent on external (environmental) parameters u = (u1, . . . , um) and smallpositive parameters ε, µ 1. Here

Λ(u) := diag(λ1(u), . . . , λn(u)), λj(·) ∈ C∞(Rm; R),

Ω20(u) := diag(ω2

01(u), . . . , ω20n(u)), ω0j(·) ∈ C∞(Rm; R++),

F (·) ∈ C∞(R2n+m+1; Rn)

andF (w, p, u, 0) = O(‖w‖2 + ‖p‖2), ‖w‖2 + ‖p‖2 → 0, (1.3)

where ‖ · ‖ :=√〈·, ·〉 stands for the Euclidean norm associated with the standard

dot product in the coordinates vector space. Hereafter, we will also use a norm | · |

EJDE-2016/233 DYNAMICAL BIFURCATION IN COUPLED OSCILLATORS 3

defined as the sum of the absolute values of vector components. We study the casewhere the slow evolution of parameters u is governed by the system

u = µG(w, w, u, µ) (1.4)

where G(·) ∈ C∞(R2n+m+1; Rm). Thus, the parameter µ plays the role of theso-called slowness parameter. When µ = 0, the system

w + Ω20(u)w = 2εΛ(u)w + F (w, w, u, 0), u = 0 (1.5)

has an invariant surface given by the equations w = w = 0 and the parameter εis responsible for the oscillations damping rate (oscillations growth rate) of vari-ables w, w near this surface for those u that belong to the stability zone where allλj(u) < 0 (complete instability zone where all λj(u) > 0). The presence of twosmall parameters in fast-slow systems is a rather usual case. Initially these smallparameters are completely independent, however, later we will impose a restric-tion that µ ∝ ε. All the functions involved in systems (1.2) and (1.4) may alsocontinuously depend on ε, but we will not show this explicitly.

By the terminology of [1], the equations w = w = 0 define the so-called slowsurface in the phase space R2n+m, the vector g(u) := G(0, 0, u, 0) is called the slowvelocity vector, and, in this way, we obtain the slow system on the slow surface

u = g(u).

In [23, 37, 38], there was considered the case when the slow system has a uniqueequilibrium attracting all its other trajectories. Here we study a more generalsituation allowing multiple equilibria, among which are stable, hyperbolic andcompletely unstable ones, but require the slow vector field to be gradient-like.This means that there exists a Morse function V (·) ∈ C2(Rm; R+) such thatVg(u) := 〈∇V (u), g(u)〉 < 0 for any non-stationary point u of V (·). We findadditional conditions under which a neighborhood of the slow surface is forwardinvariant under the semi-flow of system (1.2)–(1.4) and the set of all ω-limit pointscontained in this neighborhood consists of equilibria and invariant tori of all di-mensions less than or equal to n. We show that a typical forward semi-trajectorystarting at (w0, w0, u0), where u0 belongs to the instability zone of the fast sys-tem (1.5), demonstrates the following transient process: while the slow componentu(t) is far from the stationary points of the Morse function V , the fast component(w(t), w(t)) exhibits damping oscillations; afterwards, this component eventuallyenters and stays in a small neighborhood of some stationary point, and the oscil-lation amplitude of the fast component begins to increase. Since the trajectory isattracted by an invariant torus, eventually a multi-frequency oscillatory regime isestablished. Such behavior can be naturally interpreted as the dynamical bifurca-tion of multi-frequency oscillations.

In fact, we will also be able to categorize the solutions by their ultimate behaviornear the slow surface. It will be shown that in a small neighborhood of the slowsurface most of the system’s trajectories, in terms of the Lebesgue measure, areattracted to trajectories on the stable n-dimensional invariant torus, while the restones lie on the stable manifolds of hyperbolic tori of dimension less than n.

The current article is organized as follows. Section 2 provides the key hypothesesregarding system (1.2)–(1.4) and the statement of the main theorem. Then, insection 3 we introduce auxiliary lemmas, which enable us to state in section 4certain preliminary results on the system’s dynamics near its slow surface, and,


consequently, describe the solutions behavior and classify them in sections 5, 6.After that, we provide an example depicting oscillation excitation in a circuit of twocoupled oscillators which have components with temperature dependent properties.Finally, the paper ends with an addendum containing information on the normalforms method for systems with slowly varying parameters.

2. Main theorem

Let us describe the conditions imposed on system (1.2)–(1.4)We will assume that the slow vector field u 7→ g(u) := G(0, 0, u, 0) satisfy the

following conditions(H1) there exists a Morse function V (·) ∈ C2(Rm; R+) with the properties:

(1) V (·) has a non-empty finite set of stationary points;(2) V (u)→ +∞ when ‖u‖ → ∞;(3) Vg(u) := 〈∇V (u), g(u)〉 < 0 for any non-stationary point u of V (·);(4) the Hesse matrix ∂2

∂u2 Vg(u) is negative definite at any stationary pointof V (·).

Then, according to this hypothesis, any level set of V (·) is compact and if V ∗ > 0is sufficiently large, then the sub-level set

V := V −1([0, V ∗)) = u ∈ Rn : V (u) < V ∗

contains the setW := u ∈ Rm : ∇V (u) = 0

of all stationary points of V (·). Moreover, there exist such ν∗ ≥ ν∗ > 0 and δ > 0,that for any stationary point u∗ the following inequalities hold

−ν∗‖u− u∗‖2 < Vg(u) ≤ −ν∗‖u− u∗‖2, ‖∇V (u)‖ ≤ ν∗‖u− u∗‖ (2.1)

for all u : ‖u−u∗‖ ≤ δ, and δ-neighborhoods of any two points ofW do not intersect.Obviously,W thereby coincides with the set of all singular points of the vector fieldg.

Now, for such a number V ∗, let us adopt certain non-resonant hypotheses whichare necessary for construction of the system’s normal form in the whole domain V,as well as in a vicinity of the set W.

(H2) if u ∈ cl(V), then

ω0i(u) 6= ω0j(u), ω0i(u) 6= ω0j(u) + ω0k(u), ω0i(u) 6= ω0j(u) + ω0k(u)± ω0l(u)

for all i, j, k, l ∈ 1, 2, . . . , n.(H3) there exists such a number N ≥ 5, that for any u∗ ∈ W the equality

n∑j=1

(qj − qj+n)ω0j(u∗) = σω0i(u∗),

where σ ∈ 0, 1, i ∈ 1, . . . , n, q = (q1, . . . , q2n) ∈ Z2n+ , 4 ≤

∑2nj=1 qj ≤ N ,

is valid iff qi = qi+n + σ and qj = qj+n for all j ∈ 1, . . . , n \ i.Furthermore, we may assume that for all ε ∈ [0, ε0], with ε0 > 0 being small

enough, and for all u ∈ cl(V) the frequencies

ωj(u, ε) :=√ω2

0j(u)− (ελj(u))2 = ω0j(u) +O(ε2), j ∈ 1, . . . , n.


are correctly defined and satisfy hypotheses (H2), (H3) where each ω0j(u) is re-placed with ωj(u, ε). Hereafter, to simplify our notations, we will omit explicitdependencies of functions on ε, as long as it does not lead to confusion. Thus, wewill use the notation ωj(u) instead of ωj(u, ε) and so on.

Proceeding to the new variables w, w 7→ x = (x1, . . . , x2n) by means of a substi-tution

wj = x2j−1, wj = ελj(u)x2j−1 + ωj(u)x2j , j ∈ 1, . . . , n, (2.2)

we come to an equivalent system

x = J(u)x+ F (x, u, µ), u = µG(x, u, µ), (2.3)

where

J(u) := diag[(

ελ1(u) ω1(u)−ω1(u) ελ1(u)

), . . . ,

(ελn(u) ωn(u)−ωn(u) ελn(u)

)],

G(x, u, µ) := G(w, w, u, µ)∣∣w,w 7→x,

F (x, u, µ) = (F1(x, u, µ), . . . , F2n(x, u, µ)),

F2j−1(x, u, µ) ≡ 0,

F2j(x, u, µ) :=1

ωj(u)F (w, w, u, µ)

∣∣w,w 7→x − µε

m∑i=1

∂λj(u)∂ui

Gi(x, u, µ)x2j−1

− µm∑i=1

∂ωj(u)∂ui

Gi(x, u, µ)x2j .

In view of (1.3), when µ = 0, system (2.3) has a slow invariant manifold ofequilibria M0 defined by the equation x = 0. Alike static bifurcation theory [7, 8],we will study the behavior of system (2.3) in a neighborhood of this manifold. Andto do so, our first step will be finding conditions that guarantee the forward semi-invariance of such a neighborhood. This can be achieved by transforming the N -jetof system (2.3) to the normal form with respect to variables x.

Let DN ⊂ Rm denote a domain (or a collection of domains), such that for a fixednatural N and for any k ∈ 1, . . . , n and σ ∈ 0, 1 the equality

minu∈clDN

∣∣ n∑l=1

(ql − ql+n − σδkl)ω0l(u)∣∣ = 0,

where q ∈ Z2n+ , 1 ≤ |q| :=

∑2nk=1 qk ≤ N , fulfills iff

ql − ql+n − σδkl = 0 ∀l ∈ 1, . . . , n.

Remark 2.1. Hypothesis (H2) guarantees that D3 = V, and under hypothesis(H3) the domain DN is non-empty and contains some neighborhood of the set W.

If we introduce a vector of complex coordinates z = (z1, . . . , zn) ∈ Cn andnotations

−→|z| := (|z1|, . . . , |z|n),

−→|z|2 := (|z1|2, . . . , |z|2n),

−→|z|p :=

n∏k=1

|zk|pk ,

where p := (p1, . . . , pn) ∈ Zn+, then, as it is shown in the Addendum, for allsufficiently small µ and ε there exists a smoothly diffeomorphic change of variables


(x, u) 7→ (Re z, Im z, v) which for v ∈ DN transforms system (2.3) into

zk =[ελk(v) + iωk(v) +

∑3≤2|p|+1≤N

h0,k,p(v)−→|z|2p

]zk +

P∑j=1

µjηj,k(v)zk

+P∑j=1

µj[ ∑

3≤2|p|+1≤N

hj,k,p(v)−→|z|2p

]zk +O(‖z‖N+1 + µP+1),

k ∈ 1, . . . , n,

v = µ[g(v) +

∑2≤2|p|≤N

g0,p(v)−→|z|2p +

P∑j=1

µj∑

0≤2|p|≤N

gj,p(v)−→|z|2p

]+ µO(‖z‖N+1 + µP+1).

(2.4)

Here P ≥ N/2 is an arbitrary fixed natural number, ηj,k(·), hj,k,p(·) are smoothcomplex-valued functions in DN and gj,p(·) are smooth Cn-valued functions inDN . Besides that, all these functions smoothly depend on the parameter ε. Theremainder terms are smooth in the sense of real calculus on the set

‖z‖ < δ, v ∈ DN , µ ∈ [0, µ0], ε ∈ [0, ε0]

with sufficiently small positive numbers δ, µ0 and ε0, and are uniform with respectto v ∈ DN and ε ∈ [0, ε0].

Further, we will also denote

λ(v) = (λ1(v), . . . , λn(v)), A(v) = akl(v)nk,l=1,

akl(v) := −Reh0,k,el(v), bkl(v) := −Imh0,k,el(v),

where el ∈ Zn+ is a vector having its l-th component equal 1 and all other equal0. As we will see later, the functions λ(v) and A(v) play the key role in emergenceof the bifurcation phenomenon, which is why we require them to satisfy additionalconstraints.

(H4) the symmetrical part of the matrix A(v) is positive definite on cl(V), andall non-diagonal elements aij(v), i 6= j, are non-positive at any stationarypoint v ∈ W.

(H5) the set V admits representation as a union of three nonempty sets

V+ := v ∈ V : λj(v) > 0 ∀j ∈ 1, . . . , n,V− := v ∈ V : λj(v) < 0 ∀j ∈ 1, . . . , n,

V∗ = V\ [V+ ∪ V−] ,

and each function λj(·), j ∈ 1, . . . , n, is positive at any stationary pointof V (·).

Note, that W ⊂ V+ and for system (2.3) with µ = 0 the submanifold (x, u) ∈M0 : u ∈ V− is a local attractor, while (x, u) ∈M0 : u ∈ V− is a local repeller.

Let us fix sufficiently small κ > 0 in such a way that

Vκ− := v ∈ V− : λj(v) ≤ −κ ∀j ∈ 1, . . . , n 6= ∅.Now we are in a position to state our main result.

Theorem 2.2. There exist ρ0 > 0, ς0 > 0, and for any ς∗ ∈ (0, ς0) there is ε0 > 0,such that once ε ∈ (0, ε0) and µ ∈ (ς∗ε, ς0ε), the following statements are true:


(1) system (2.4) generates a forward semi-flow on the set

A := (z, v) ∈ Cn × Rm : ‖z‖ ≤ ρ0, v ∈ V;

(2) if R+ 3 t 7→ (z(t), u(t)) is a solution of system (2.4), such that (z(0), v(0)) ∈A and (t0, t1) ∈ R+ is an interval, such that ‖z(t)‖ > µP and v(t) ∈ Vκ− forall t ∈ (t0, t1), then

‖z(t)‖ ≤ e−εκ(t−t0)/2‖z(t0)‖ ∀t ∈ (t0, t1);

(3) to any point of the setW, one can put into correspondence a finite collectionof invariant tori belonging to A, and each such collection contains tori ofall dimensions from 1 to n; in addition, any torus of a dimension less thenn is truly hyperbolic, while any n-dimensional torus corresponding to a localminimum of the Morse function V (·) is a local attractor of system (2.4);

(4) any non-equilibrium forward semi-trajectory of this system lying in A is at-tracted by one of the invariant tori, and those trajectories that are attractedby n-dimensional tori form the set of the full Lebesgue measure in A;

(5) each forward semi-trajectory approaching the n-dimensional invariant torusis attracted by a forward semi-trajectory lying on this torus.

The rest of this article will be devoted to proving and illustrating this theorem.

3. Auxiliary lemmas

Lemma 3.1. For δ > 0, set Bdδ := x ∈ Rd : ‖x‖ ≤ δ. Suppose that there existsa Morse function W (·) ∈ C2(Bdδ ; R) having a unique stationary point x∗ = 0 and avector field f(·) ∈ C1(Bdδ ; Rd), such that

〈∇W (x), f(x)〉 ≤ −θ‖x‖2, ‖f ′(x)‖ ≤ θL ∀x ∈ Bdδ (0),

where L and θ are some positive constants. Define

K := max‖x‖≤δ

‖W ′′(x)‖

and let M and ε be arbitrary positive numbers satisfying

M ≥M∗(K,L) :=1 +KL+

√1 + 2K3L3

L, Mε < δ. (3.1)

Then for any f1(·) ∈ C1([τ0,∞)× Bdδ ; Bdθ ) and any x0 ∈ Bdδ a solution x(t) of theinitial problem

x = f(x) + εf1(t, x), x(τ0) = x0 (3.2)

meets the following alternative: either there exists such τ∗ > τ0, that ‖x(τ∗)‖ = δ,or there exists such τ∗ ≥ τ0, that ‖x(t)‖ < Mε for all t > τ∗.

Additionally, if 0 < ε < δ/N(K,L,M), where

N(K,L,M) :=1 +KL+

√[1 + (K −M)L]2 + 2KL3M2

L> M, (3.3)

and if the first scenario takes place, but at some instant of time the solution belongsto BdMε, then

W (x(τ∗)) < W (0)− K(Mε)2

2≤ minW (x) : ‖x‖ = Mε.


Furthermore, in the case when the stationary point x∗ = 0 is elliptic, numbersβ ∈ (0, δ) and ε > 0 are such, that

maxW (x) : ‖x‖ = β < minW (x) : ‖x‖ = δ, 0 < ε < β/K, (3.4)

and the solution of (3.2) at some moment of time belongs to Bdβ, then the secondscenario fulfills.

Proof. Since ∇W (0) = 0, it is true that ‖∇W (x)‖ ≤ K‖x‖ and

〈∇W (x), f(x) + εf1(t, x)〉 ≤ θ [−‖x‖+ εK] ‖x‖ ∀x ∈ Bdδ .

Besides,

maxW (x) : ‖x‖ = % ≤W (0) +K%2

2, minW (x) : ‖x‖ = % ≥W (0)− K%2

2for any % ∈ (0, δ]. As the Hessian of W (·) at x = 0 is non-degenerate, we havef(0) = 0 and ‖f(x)‖ ≤ θL‖x‖. Hence,

‖f(x) + εf1(t, x)‖ ≤ θ(L‖x‖+ ε) ∀x ∈ Bdδ .

Now let us demonstrate how to choose M . At first, we will require only thatM ≥ K and Mε < δ. Take an arbitrary % ∈ (Kε,Mε). If the moment τ∗ does notexists, i. e. ‖x(t)‖ < δ for all t ≥ τ0, then the function W (x(t)) strictly decreasesuntil x(t) reaches the sphere ‖x‖ = % at an instant of time τ∗ ≥ τ0. The momentτ∗ necessarily exists, since otherwise W (x(t)) would decrease unboundedly, whichis impossible.

Suppose that x(t) reaches the sphere ‖x‖ = Mε after the moment τ∗. Then thereexist τ2 > τ∗ and τ1 ∈ [τ∗, τ2), such that

‖x(τ1)‖ = %, % < ‖x(t)‖ < Mε ∀t ∈ (τ1, τ2), ‖x(τ2)‖ = Mε.

For t ∈ [τ1, τ2], we have

d‖x(t)‖dt

≤ 〈x, f(x) + εf1(t, x)〉‖x‖

∣∣x=x(t)

≤ θ(L‖x(t)‖+ ε),

which impliesd‖x(t)‖/dt

θ(L‖x(t)‖+ ε)≤ 1

and

W (x(τ2))−W (x(τ1)) =∫ τ2

τ1

〈∇W (x), f(x) + εf1(t, x)〉∣∣x=x(t)

dt

≤∫ τ2

τ1

θ [−‖x(t)‖+Kε] ‖x(t)‖ d‖x(t)‖/dtθ(L‖x(t)‖+ ε)

dt

=∫ Mε

%

(−s+Kε)sLs+ ε

ds.

Taking into account that W (x(τ1)) ≤W (0) +K%2/2 and making % tend to Kε, weobtain the estimate

W (x(τ2)) ≤W (0) +K3ε2

2+[− s2

2L+ε(1 +KL)s

L2− ε2(1 +KL)

L3lnLs+ ε

L

]Mε

Kε

< W (0) +K3ε2

2− ε2

2L[M2 − 2(1 +KL)M

L+

2(1 +KL)KL

−K2].


If we introduce a quadratic polynomial of variable ξ,

P(ξ; ε, η) := ξ2 − 2(1 +KL)εL

ξ +2(1 +KL)ηε2

L− (1 + 2KL)η2ε2,

where ε and η are positive parameters, one can verify that M∗(K,L) is the greatestroot of P(ξ; 1,K). Thus, for any M ≥ M∗(K,L) and any ε < δ/M , we obtainP(Mε; ε,K) = ε2P(M ; 1,K) ≥ 0. It means that

M2 − 2(1 +KL)ML

+2(1 +KL)K

L−K2 ≥ 2K3L,

W (x(τ2)) < W (0)− K3ε2

2≤ minW (x) : ‖x‖ = Kε.

Hence, the function W (x(t)) keeps on decreasing and satisfies for t > τ2 the inequal-ity W (x(t)) < W (x(τ2)). This means that x(t) never reaches the sphere ‖x‖ = Kε,and moreover,

inft≥τ2

[‖x(t)‖ −Kε] > 0.

As a consequence, W (x(t)) → −∞ as t → ∞, and we come to a contradiction.Therefore, such a choice of M and ε guarantees the validity of the inequality‖x(t)‖ < Mε for all t > τ∗.

In a similar way, one can show that if ‖x(τ ′)‖ ≤ Mε, but there exists τ∗ > τ ′,such that ‖x(τ∗)‖ = δ, then

W (x(τ∗))

≤W (0) +KM2ε2

2−[ s2

2L− ε(1 +KL)s

L2

ε2(1 +KL)L3

lnLs+ ε

L

]δMε

≤W (0) +KM2ε2

2− 1

2L[δ2 − 2(1 +KL)ε

Lδ +

2(1 +KL)Mε2

L− (Mε)2

].

Since N(K,L,M) is the greatest root of P(ξ; 1,M), once 0 < N(K,L,M)ε < δ, wehave P(δ; ε,M) ≥ 0 and

W (x(τ∗)) < W (0)− K(Mε)2

2≤ minW (x) : ‖x‖ = Mε.

Finally, if the point 0 is elliptic and inequalities (3.4) are fulfill, then

〈∇W (x), f(x) + εf1(t, x)〉 ≤ [−θ‖x‖+ εK] ‖x‖ < 0, (3.5)

as soon as β ≤ ‖x‖ ≤ δ. Let the solution belong to Bdβ at some moment of time.If by reasoning ad absurdum we supposed that there existed such τ∗ > τ0, that‖x(τ∗)‖ = δ, then there would exist τ ′′ < τ∗, such that

‖x(τ ′′)‖ = β, β < ‖x(t)‖ < δ ∀t ∈ (τ ′′, τ∗).

Thereby, W (x(τ ′′)) ≤ max‖x‖=βW (x) < min‖x‖=δW (x(τ∗)), which is impossible,since (3.5) yields that the function W (x(t)) is decreasing on (τ ′, τ∗).

Lemma 3.2. Let D ⊂ Rd be a bounded domain with a C2-boundary, D := cl(D)and let W (·) ∈ C2(D; R) be a Morse function with a finite set of stationary pointsW ⊂ D. Define

K := maxx∈D‖∇W (x)‖, ‖W ′′(x)‖

and choose sufficiently small δ > 0 and β ∈ (0, δ) that meet the following require-ments:


(1) the δ-neighborhood of W belongs to D;(2) for any x′∗, x

′′∗ ∈ W, such that W (x′∗) > W (x′′∗) the following inequality

holds:

minW (x) : ‖x− x′∗‖ = δ > maxW (x) : ‖x− x′′∗‖ = δ;

(3) for any elliptic point x∗ ∈W there holds the inequality

maxW (x) : ‖x− x∗‖ = β < minW (x) : ‖x− x∗‖ = δ.

Also, suppose that there is such a vector field f(·) ∈ C1(D; Rd), that for somepositive constants θ, L such inequalities fulfill:

〈∇W (x), f(x)〉 < −θδ2 ∀x ∈ D : miny∈W‖x− y‖ > δ,

〈∇W (x), f(x)〉 ≤ −θ‖x− y‖2, ‖f ′(x)‖ ≤ θL ∀y ∈W,∀x : ‖x− y‖ ≤ δ.

Then, with the corresponding functions defined by formulae (3.1) and (3.3), for anyM ≥M∗(K,L) and all ε ∈ (0, ε0(K,L,M)), where

ε0(K,L,M) := minδ2

K,

β

N(K,L,M),

the following assertion is correct. If f1(·) ∈ C1([τ0,∞) × D; Bdθ ), then for anysuch x0 ∈ D, that the corresponding solution x(·) of initial problem (3.2) is definedon [τ0,∞) and takes values in D, there exist x∗ ∈ W and t∗ > τ0, such that‖x(t)− x∗‖ < Mε for all t > t∗.

Proof. Under the conditions of this lemma, we have

〈∇W (x), f(x) + εf1(t, x)〉 < 0 ∀x : miny∈W‖x− y‖ ≥Mε. (3.6)

Therefore, if the solution x(·) is defined on [τ0,∞) and takes values in D, then thereexist x1

∗ ∈ W and t1 ≥ τ0, such that ‖x(t1) − x1∗‖ < Mε. Indeed, otherwise, the

function W (x(t)) would decrease unboundedly when t→∞, which is impossible.By Lemma 3.1, the solution x(·) meets the following alternative: either for all

t ≥ t1 we have ‖x(t)−x∗‖ ≤Mε, or there exists t2 > t1, such that ‖x(t2)−x1∗‖ = δ

and

W (x(t2)) < W (x1∗)−

K(Mε)2

2≤ minW (x) : ‖x− x1

∗‖ = Mε.

The first case always takes place if x1∗ is elliptic. In the second one, on account of

choice of δ and (3.6), there exist such x2∗ ∈W and t3 > t2, that ‖x(t3)−x2

∗‖ < Mε,and W (x2

∗) < W (x1∗). Now, it is clear that eventually the solution enters and then

never leaves an Mε-neighborhood of some point x∗ ∈W.

For the sake of completeness of our exposition, let us represent the followingresult from the theory of non-negative invertible matrices.

Lemma 3.3. Let a real matrix P = pijdi,j=1 have such properties:

(1) the matrix P + P> is positive definite;(2) pij ≤ 0 for any i, j ∈ 1, . . . , d, i 6= j.

Then for any vector y = (y1, . . . , yd) with the positive elements each component ofthe vector x := P−1y satisfies the inequalities

xi ≥yipii, i ∈ 1, . . . , d, (3.7)


|x| ≤ max1≤i≤d yip+

, (3.8)

where p+ := min〈Pξ, ξ〉 : ξ ∈ Rd+, |ξ| = 1,

Proof. The system x = −Px, in which P has the property 1 is asymptoticallystable. Hence, all eigenvalues of −P have negative real parts. By [16, Theorem 5,XIII], all of the main minors of P are positive. Thus, according to [36, Theorem6.3] the matrix P−1 has non-negative elements, and any row of this matrix containsat least one positive element. Obviously, then x := P−1y ∈ Rd++, once y ∈ Rd++.Moreover, since

d∑j=1,j 6=i

pijxj ≥ 0,

the components xi satisfy (3.7), and the inequalities

p+|x|2 ≤ 〈Px, x〉 = 〈y, x〉 ≤ max1≤i≤d

yi|x|

yield (3.8).

4. Preliminary results on the behavior of the normalized system

Having introduced the aforementioned lemmas, we may proceed to the investi-gation of system (2.4) dynamics. This section provides the general description ofthe solutions behavior and suggests a way to classify them. Later we will refine thisinformation. Define

a+ := min〈A(v)r, r〉 : ∀r ∈ Rn+, |r| = 1, v ∈ cl(V),λ+ := max〈λ(v), r〉 : ∀r ∈ Rn+, |r| = 1, v ∈ cl(V).

Proposition 4.1. Assume hypotheses (H1), (H2), (H4) to be true and N = 3.Then there exist a sufficiently small ρ0 > 0 and sufficiently large ρ∗ > 0, R∗ > 0,such that for any ρ > ρ∗, R > R∗ one can choose ε0 > 0 in such a way, that onceε ∈ (0, ε0] and µ ∈ [0, ε], system (2.4) generates a forward semi-flow on the setA := (z, v) ∈ Cn×Rm : ‖z‖ ≤ ρ0, v ∈ V. Furthermore, for any solution (z(t), v(t))of (2.4) with the initial values (z(0), v(0)) ∈ A, there exist such a stationary pointv∗ of V (·) and an instant of time t∗, that

‖z(t)‖ < ρ√ε < ρ0, ‖v(t)− v∗‖ < Rε ∀t ≥ t∗.

Proof. There is a constant c1 > 0, such that if ‖z‖ ≤ ρ0, then a quadratic form‖z‖2 =

∑nk=1 |zk|2 and the function V (·) admit the following estimates for their

directional derivatives along the vector field of system (2.4)

‖z‖2∣∣′(2.4)≤ 2ε

n∑k=1

λk(v)|zk|2 − 2n∑

k,l=1

akl(v)|zk|2|zl|2

+ c1(µ‖z‖2 + ‖z‖5 + ‖z‖µP+1)

≤[(2ελ+ + c1µ)‖z‖ − (2a+ − c1ρ0)‖z‖3 + c1µ

P+1]‖z‖,

V (v)∣∣′(2.4)≤ µ

[Vg(v) + c1‖∇V (v)‖(‖z‖2 + µ)

]. (4.1)

It is easily seen that one can choose the positive numbers ρ0 < 2a+/c1, and ρ∗ > 0in such a way, that for any ρ > ρ∗, µ ∈ [0, ε], v ∈ cl(V), ε ∈ (0, ε1], where


ε1 := min1, ρ20/ρ

2, µ0, the inequality

‖z‖2∣∣′(2.4)

< 0,

holds as long as z satisfies ρ√ε ≤ ‖z‖ ≤ ρ0. It means that A is forward semi-

invariant and, moreover, there exists t0 > 0, such that ‖z(t)‖ < ρ√ε for all t > t0.

Now we can regard v(t) as a solution of a system v = µ[g(v) + g(t, v, ε, µ)] definedon [t0,∞)× cl(V) and obtained from the corresponding sub-system of (2.4) via thesubstitution z = z(t). Obviously, there exists such a constant c2 > 0, that

‖g(t, v, ε, µ)‖ ≤ c2ε ∀(t, v, ε, µ) ∈ [t0,∞)× cl(V)× (0, ε1]× [0, ε].

On account of (H1), (2.1) and (4.1), after an appropriate additional correctionof δ in (2.1), the final part of the proof follows from Lemma 3.2 in the case whenf = µg, f1 = µεg, W = V , ε = εc2, θ ∝ µ. In particular, if we find M∗ and ε0 fromthe lemma, we can set R∗ = M∗, R = M and ε0 = minε0/c2, ε1.

Corollary 4.2. Let 0 < µ < εκ/(2c1) and let (t0, t1) ∈ R+ be such an interval,that

µP < ‖z(t)‖ < ρ0, v(t) ∈ Vκ− ∀t ∈ (t0, t1)

Then ‖z(t)‖ ≤ e−εκ(t−t0)/2‖z(t0)‖ for all t ∈ (t0, t1).

In fact, for µP ≤ ‖z‖ ≤ ρ0, v ∈ Vκ− and 0 < µ < εκ/(2c1), in the same way as inthe proof of Proposition 4.1, we obtain the inequality

‖z‖2∣∣′(2.4)≤[(−2εκ+ c1µ)‖z‖ − (2a+ − c1ρ0)‖z‖3 + c1µ

P+1]‖z‖

≤[− 3εκ

2‖z‖+

εκ

2µP]‖z‖ ≤ −εκ

2‖z‖2.

The above corollary proves the statement (2) of the main theorem.Hypothesis (H3) and Proposition 4.1 allow us to focus on system (2.4) defined

on the set(r, v) : ‖z‖ < ρ

√ε, ‖v − v∗‖ < Rε.

Hereafter, we will require the numbers ρ and R to be large enough.Without loss of generality we may suppose that v∗ = 0 in Proposition 4.1. Then,

having applied the scaling z 7→√εz, v 7→ εv to (2.4), we obtain the system

zk =[iωk(εv) + ελk(εv)− ε

n∑l=1

(akl(εv) + ibkl(εv))|zl|2]zk

+∑

5≤2|p|+1≤N

ε|p|h0,k,p(εv)−→|z|2pzk +

P∑j=1

µjηj,k(εv)zk

+P∑j=1

µj[ ∑

3≤2|p|+1≤N

ε|p|hj,k,p(εv)−→|z|2p

]zk +O

(εN/2‖z‖N+1 + µP

),

k ∈ 1, . . . , n,

v =µ

ε

[g(εv) +

∑2≤2|p|≤N

ε|p|g0,p(εv)−→|z|2p

]

+µ

ε

P∑j=1

µj∑

0≤2|p|≤N

ε|p|gj,p(εv)−→|z|2p +O

(µε(N−1)/2‖z‖N+1 + µP+2/ε

)

(4.2)


defined on the set

(z, v, ε, µ) : ‖z‖ ≤ ρ, ‖v‖ ≤ R, ε ∈ (0, ε0], µ ∈ [0, ε].

In this system we constrain the parameter µ to be µ = ες with ς being an arbitraryfixed number satisfying

0 < ς ≤ ς0 :=12

min1≤j≤n

λj(0)|η1,j(0)|

.

Note that such a condition ensures the validity of the inequality

αk := λk(0) + ς Re η1,k(0) ≥ λk(0)/2 > 0 ∀k ∈ 1, . . . , n.

Now, if we recall the earlier imposed condition P ≥ N/2 and if for a vectorx = (x1, . . . , xd) we define D[x] := diag(x1, . . . , xd), then system (4.2) can bepresented in the form

z = D[i(ω0 + ε(β + Ω′v −B−→|z|2))]z

+ [ε(α−A−→|z|2) + ε2h(

−→|z|2, v)]z +O(εN/2),

(4.3)

v = ες[Γv + Υ−→|z|2 + εg(

−→|z|2, v) +O(ε(N+1)/2)] (4.4)

where, for the sake of notations simplicity, we assign

ω0 := (ω01(0), . . . ω0n(0)), η1 := (η1,1(0), . . . , η1,n(0)), α := λ(0) + ς Re η1(0),

β := ζ Im η1(0), A :=akl(0)

nk,l=1

, B :=bkl(0)

nk,l=1

,

Ω′ :=∂ω0k(0)

∂vj

nk=1,

Γ := g′(0), Υx :=∑|p|=1

g0,p(0)xp.

The definitions of the remainder terms h(·) and g(·) inside the square brackets inthe right-hand sides of (4.3), (4.4) is obvious. Recall that we agreed not to mentionε directly as functions arguments.

Proposition 4.3. For all ε ∈ (0, ε0], with ε0 > 0 being sufficiently small, sys-tem (4.3)–(4.4) has an equilibrium (z0, v0) = (z0(ε), v0(ε)), such that

z0(ε) = O(εN/2), v0(ε) = O(ε). (4.5)

If (z(t), v(t)) is a solution of (4.3)–(4.4), such that ‖z(t)‖ <√ε and ‖v(t)‖ < R

for all t > 0, thenlimt→∞

[‖z(t)− z0‖+ ‖v(t)− v0‖

]= 0, (4.6)

and the set of all such solutions forms a manifold whose real dimension equals thenumber of eigenvalues of Γ with negative real parts. In the case when all eigenval-ues of Γ have positive real parts the only solution with the stated property is theequilibrium (z0, v0).

Proof. The existence of the equilibrium (z0(ε), v0(ε)) satisfying (4.5) directly fol-lows from the implicit function theorem. If (z(t), v(t)) is a solution of (4.3)–(4.4),such that ‖z(t)‖ <

√ε and ‖v(t)‖ < R for all t > 0, then the functions

w(t) := exp[− i∫ t

0

D[(ω0 + ε(Ω′v(s)−B

−−−→|z(s)|2)

)]ds]z(t)


and v(t) satisfy a system of the form

w = D[ε(α−A

−→|w|2) + ε2h(

−→|w|2, v)

]w +O(εN/2), (4.7)

v = ες[Γv + Υ

−→|w|2 + εg(

−→|w|2, v) +O(ε(N+1)/2)

], (4.8)

which also has a solution

w = w0(t) := exp[− i∫ t

0

D[(ω0 + ε(Ω′v(s)−B

−−−→|z(s)|2))

]ds]z0,

v = v0, t ≥ 0.

Hence, the realification of system (4.7)–(4.8) has a pair of solutions

ξ(t) := (Rew(t), Imw(t), v(t)), ξ0(t) := (Rew0(t), Imw0(t), v0).

One can consider the difference η(t) := ξ(t)− ξ0(t) to be a bounded solution fort ≥ 0 of the linear system η = ε[A+A1(t; ε)]η where

A = diag[D[α], D[α], ςΓ],

and supt≥0 ‖A1(t; ε)‖ = O(√ε). Such a system is hyperbolic on [0,∞) if ε is small

enough, and each of its bounded solutions approaches zero as t → ∞. This yields(4.6).

Next, it is not hard to see that at (z0, v0) the Jacobi matrix of the right-handside of system (4.3)–(4.4) realification has the form

±iω0k + ε(αk ± iβk) + o(ε), k ∈ 1, . . . , n, εςγj + o(ε), j ∈ 1, . . . ,mwhere γ1, . . . , γm are the eigenvalues of Γ counted according to multiplicities. Allof these numbers have non-zero real parts. It is well known that in this case theset of all solutions of the realification of (4.3)–(4.4) which approach the equilibriumξ0(ε) forms the so-called stable manifold, whose dimension equals the number ofeigenvalues of A(ε) with negative real parts (see, e. g. [21]). In the case when allof the eigenvalues of Γ have positive real parts, the equilibrium (Re z0, Im z0, v0) iscompletely unstable.

Proposition 4.4. There exists a constant c3 > 0, such that for sufficiently smallε0 > 0 and for any ε ∈ (0, ε0] the following assertion is valid. If (z(t), v(t)) is asolution of (4.3)–(4.4), such that ‖z(0)‖ ≤ ρ, |zk(0)| ≥

√c3εN−2 for some k ∈

1, . . . , n and ‖v(t)‖ < R for all t > 0, then

|zk(t)| >√c3εN−2, ‖z(t)‖ < ρ ∀t > 0,

and there exists such t∗ > 0, that

|zk(t)| >√

αk2akk

, ‖z(t)‖ <√

2 max1≤k≤n αka0

+

∀t > t∗,

wherea0

+ := min〈Ar, r〉 : r ∈ Rn+, |r| = 1.

Proof. It is sufficient to note that there is such c4 > 0, that for sufficiently smallε0 > 0 and for any ε ∈ (0, ε0] the following inequalities are satisfied.

‖z‖2∣∣′(4.3)≤ 2ε〈

−→|z|2, α−A

−→|z|2〉+ c4(ε2‖z‖2 + εN/2‖z‖)

≤ 2ε[

max1≤k≤n

αk − a0+‖z‖2 + c4ε

]‖z‖2 + c4ε

N/2‖z‖ < 0


if 2 max1≤k≤n αk/a0+ ≤ ‖z‖2 ≤ ρ2, and, in view of (H4),

|zk|2∣∣′(4.3)≥ 2ε|zk|2

[αk − akk|zk|2 − c4ε− c4ε(N−2)/2/|zk|

]> 0

if c3εN−2 ≤ |zk|2 ≤ αk/(2akk), where c3 is a sufficiently large positive constant.

Definition 4.5. Let i1, . . . , is be an ordered collection of distinct natural num-bers not exceeding n. We will say that a solution (z(t), v(t)) of (4.3)–(4.4) is oftype (i1, . . . , is) if

‖v(t)‖ < R, |zk(t)| < ε ∀k 6∈ i1, . . . , is, ∀t > 0

and there exists such a moment of time t∗ > 0, that

|zk(t)| >√

αk2akk

∀k ∈ i1, . . . , is, ‖z(t)‖ <√

2 max1≤k≤n αka0

+

∀t > t∗.

As a consequence of Propositions 4.3 and 4.4, we obtain the following result.

Proposition 4.6. Let (z(t), v(t)) be a solution of (4.3)–(4.4), such that ‖z(0)‖ ≤ ρand ‖v(t)‖ ≤ R for all t ≥ 0. If ε0 is sufficiently small and ε ∈ (0, ε0], then eitherthis solution tends to the equilibrium (z0, v0) as t→∞, or it is of type (i1, . . . , is)for some s ∈ 1, . . . , n.

Proof. If ‖z(t)‖ <√ε for all t > 0, then (z(t), v(t)) is a solution of (4.3)–(4.4) from

Proposition 4.3, and therefore tends to (z0, v0) as t → ∞. If (z(t), v(t)) does notpossess the aforementioned property, then there exist k ∈ 1, . . . , n and t1 ≥ 0,such that |zk(t1)|2 > ε/n ≥ c3εN−2. Hence, the solution (z(t−t1), v(t−t1)) satisfiesthe conditions of Proposition 4.4. Now it becomes apparent that in such case, basingon Proposition 4.4, one can decompose a set 1, . . . , n into two ordered subsets,i1, . . . , is 3 k and j1, . . . , jn−s ⊂ 1, . . . , n \ i1, . . . , is, and choose t∗ > 0 insuch a way, that |zi(t)|2 > αi/(2aii) for all t > t∗ if i belongs to the first subset,whereas |zj(t)|2 ≤ c3ε

N−2 < ε2 for all t > 0 if j belongs to the second subset,with the latter being empty when s = n. Besides that, ‖z(t)‖ meets the imposedrequirements for all t > t∗.

5. Ultimate behavior of solutions of type (1, . . . , n)

To study the final behavior of solutions of type (1, . . . , n), introduce the polar-likecoordinates zk =

√rkeiϕk , k = 1, . . . , n, and set r = (r1, . . . , rn). This transforms

system (4.3)–(4.4) into

r = 2εD[r] [α−Ar + εa(r, v)] + εN/2D1/2[r]a(r, v, ϕ), (5.1)

v = ες[Υr + Γv + εg(r, v) + ε(N+1)/2g(r, v, ϕ)

], (5.2)

ϕ = ω0 + ε(β + Ω′v −Br) + ε2b(r, v) + εN/2D−1/2[r]b(r, v, ϕ) (5.3)

where a(r, v) = Re h(r, v), b(r, v) = Im h(r, v), and a(r, v, ϕ), b(r, v, ϕ), g(r, v, ϕ)are defined by the remainder terms of (4.3)–(4.4). On ground of Lemma 3.3 andProposition 4.1, we can consider ρ and R to be so large, that

|A−1α| ≤ max1≤k≤n αka0

+

<ρ

2, ‖Γ−1ΥA−1α‖ < R

2. (5.4)


Proposition 5.1. For sufficiently small ε0 > 0 and for any ε ∈ (0, ε0], the system

α−Ar + εa(r, v) = 0, Υr + Γv + εg(r, v) = 0

has a solution

r = r∗(ε) := A−1α+O(ε), v = v∗(ε) := −Γ−1ΥA−1α+O(ε),

such that

r∗k(ε) >2αk3akk

, k ∈ 1, . . . , n, |r∗(ε)| <3 max1≤k≤n αk

2a0+

, ‖v∗(ε)‖ <2R3.

Proof. Taking into account hypothesis (H4), Lemma 3.3 and inequalities (5.4), thedesired result follows from the implicit function theorem.

Proposition 5.2. There exist such positive numbers c5, ς0 and ε0, that for anyς ∈ (0, ς0), ε ∈ (0, ε0) the following assertion is true. If (z(t), v(t)) is a solution of

type (1, . . . , n) of system (4.3)–(4.4) and r(t) :=−−−→|z(t)|

2then there is such t∗ > 0,

that √‖r(t)− r∗(ε)‖2 + ‖v(t)− v∗(ε)‖2 <

c5ε(N−2)/2

ς∀t > t∗.

Proof. Let (r(t), v(t), ϕ(t)) represent the solution (z(t), v(t)) of type (1, . . . , n) inthe polar-like coordinates. On account of (5.1)–(5.3) and Proposition 5.1, the pair(r(t), v(t)) satisfies the system

r = 2εD[r][(−A+ εAr(r, v))(r − r∗) + εAv(r, v)(v − v∗)

]+ εN/2D1/2[r]a(r, v, ϕ(t)),

v = ες[(Υ + εGr(r, v))(r − r∗) + (Γ + εGv(r, v))(v − v∗)

+ ε(N+1)/2g(r, v, ϕ(t))],

where

Ar(r, v) :=∫ 1

0

∂a(r, v)∂r

∣∣∣ r 7→sr+(1−s)r∗v 7→sv+(1−s)v∗

ds,

Av(r, v) :=∫ 1

0

∂a(r, v)∂v

∣∣∣ r 7→sr+(1−s)r∗v 7→sv+(1−s)v∗

ds,

Gr(r, v) :=∫ 1

0

∂g(r, v)∂r

∣∣∣ r 7→sr+(1−s)r∗v 7→sv+(1−s)v∗

ds,

Gv(r, v) :=∫ 1

0

∂g(r, v)∂v

∣∣∣ r 7→sr+(1−s)r∗v 7→sv+(1−s)v∗

ds.

By Definition 4.5, there exists t∗ > 0, such that for all t > t∗ a point (r(t), v(t))belongs to the domain

D :=

(r, v) ∈ Rn+ × Rm : rk >αk

2akk∀k ∈ 1, . . . , n,

|r| < 2 max1≤k≤n αka0

+

, ‖v‖ < R,


which contains the unique equilibrium (r∗, v∗) of the system

r = 2εD[r][(−A+ εAr(r, v)

)(r − r∗) + εAv(r, v)(v − v∗)

],

v = ες[(Υ + εGr(r, v))(r − r∗) +

(Γ + εGv(r, v)

)(v − v∗)

],

(5.5)

This equilibrium is the unique stationary point of the Morse function

W (r, v) :=n∑i=1

(ri + r∗i ln(

r∗iri

)− r∗i)

+12〈V ′′(0)(v − v∗), v − v∗〉

in cl(D).The first inequality (2.1) yields 〈V ′′(0)Γv, v〉 ≤ −ν∗‖v‖2 for any v ∈ Rm. There-

fore, there is such a constant c6 > 0, that for sufficiently small ε0 and for allε ∈ (0, ε0), (r, v) ∈ cl(D) the following inequality holds.

W (r, v)|′(5.5)

≤ 2ε〈r − r∗,(−A+ εAr(r, v)

)(r − r∗) + εAv(r, v)(v − v∗)〉

+ ες〈V ′′(0)(v − v∗),(Υ + εGr(r, v)

)(r − r∗) + (Γ + εGv(r, v))(v − v∗)〉

≤ −εa0‖r − r∗‖2 + 2c6(ε2 + ες)‖r − r∗‖‖v − v∗‖ −εςν∗

2‖v − v∗‖2,

wherea0 := min〈Aζ, ζ〉 : ζ ∈ Rn, ‖r‖ = 1.

Now, observe that for sufficiently small positive ς0 and ε0, and for any ς ∈ (0, ς0)and ε ∈ (0, ε0) the smallest eigenvalue of the matrix(

a0 −c6(ε+ ς)−c6(ε+ ς) ςν∗/2

)exceeds ςκ, where

κ0 :=12· a0ν∗

2a0 + ςν∗.

Hence,W (r, v)

∣∣′(5.5)≤ −εςκ0

[‖r − r∗‖2 + ‖v − v∗‖2

],

as soon as (r, v) ∈ cl(D), ς ∈ (0, ς0) and ε ∈ (0, ε0). It only remains to applyLemma 3.2 in the case of unique stationary point of Morse function with θ ∝ εςand ε ∝ εN/2 to finish this proof.

Next, we are going to utilize results on the existence of invariant tori obtainedin [19]. To do so, we introduce a new vector variable ξ ∈ Rn+m via the formula

(r, v) = (r∗, v∗) + εσξ

with 0 < σ < 1/2, and define the block matrices

B :=(−2D[r∗]A 0

ςΥ ςΓ

), B1(εσξ) :=

(−2D[r − r∗]A 0

0 0

) ∣∣∣(r,v)=(r∗,v∗)+εσξ

,

B(εσξ) :=(

2D[r]Ar(r, v) 2D[r]Av(r, v)ςGr(r, v) ςGv(r, v)

) ∣∣∣(r,v)=(r∗,v∗)+εγξ

,

the vector functions

f(εσξ, ϕ) :=(D1/2[r]a(r, v, ϕ), ε3/2ςg(r, v, ϕ)

)∣∣(r,v)=(r∗,v∗)+εσξ

,


Φ(εσξ, ϕ) := b(r, v)∣∣(r,v)=(r∗,v∗)+εσξ

,

Φ(εσξ, ϕ) := ε(N−4)/2D−1/2[r]b(r, v, ϕ)∣∣(r,v)=(r∗,v∗)+εσξ

,

the vector ω∗ := ω0 + ε(β+ Ω′v∗−Br∗) and, finally, the (n× (n+m))-matrix Ξ :=[−B; Ω′]. After that, system (5.1)–(5.3) takes the following form in the variables(ξ, ϕ).

ξ = ε[B + B1(εσξ) + εB(εσξ)]ξ + εN/2−γ f(εσξ, ϕ), (5.6)

ϕ = ω∗ + ε1+γΞξ + ε2Φ(εσξ) + εN/2Φ(εσξ, ϕ). (5.7)

Assuming that ε is small enough for ω∗ to be positive, we hereby arrive at theperturbation problem for the trivial invariant torus ξ = 0 of the system

ξ = εBξ, ϕ = ω∗.

Note, that the quadratic forms 〈D[r∗(0)]A·, ·〉 and 〈V ′′(0)Γ·, ·〉 are positive andnegative definite respectively, which means that the eigenvalues of B have non-zeroreal parts. The Lipschitz constants for the right-hand sides of (5.6)–(5.7) are o(ε).Under such circumstances, one can easily verify that system (5.6)–(5.7) satisfies allconditions of [19, Lemma 2.1] (There is a misprint in condition (ii) on page 507:in Lip(θ, y, z; Σσ,µ; η(ε, σ, µ)) the symbol η should be replaced by λ.) accordingto which, there exists sufficiently small ε0 > 0, such that for any ε ∈ (0, ε0) sys-tem (5.6)–(5.7) has an invariant torus given by the equation ξ = ξ(ϕ; ε), whereξ(·; ε) : Tn → Rn+m is a Lipschitzian mapping, such that maxϕ∈Tn ‖ξ(ϕ; ε)‖ → 0as ε→ 0. Therefore, we have shown that system (5.1)–(5.3) has an invariant torusT nε located in D.

Also, if ξ(t) is a solution of system (5.1)–(5.3) corresponding to (r(t), v(t)), then

‖ξ(t)‖ < c5ε(N−2)/2−γ

ς∀t > t∗.

Lemma 2.3 in [19] claims that the trajectory corresponding to the solution ξ(t)belongs to the stable invariant manifold of the invariant torus. This yields theinequality

‖ξ(ϕ; ε)‖ ≤ c5ε(N−2)/2−γ

ς∀ψ ∈ Tn.

Furthermore, if N ≥ 5 and V ′′(0) is positive definite, then it was shown in [37]that not only does the trajectory corresponding to the solution from Proposition 5.2approach the torus T nε as t→∞, but it is also attracted by some trajectory on T nεwhen t→∞.

Hence, we have proved the part of statements (3)–(5) from the main theoremconcerning n-dimensional tori.

6. Ultimate behavior of solutions of type (i1, . . . , is)

Now, we will cover the case when solutions are of type (1, . . . , s). Obviously,results that we are going to obtain will be applicable to other possible types(i1, . . . , is), too.

We introduce new variables p := (p1, . . . , ps) ∈ Rs+, ϑ := (ϑ1, . . . , ϑs) ∈ Ts andζ = (ζ1, . . . , ζn−s) ∈ Cn−s via the formulae

zk =√pkeiϑk , k ∈ 1, . . . , s, zk+s = εζk, k ∈ 1, . . . , n− s.


If we denote

α′ = (α1, . . . , αs), α′′ = (αs+1, . . . , αn),

β′ = (β1, . . . , βs), β′′ = (βs+1, . . . , βn),

ω′0 = (ω01, . . . , ω0s), ω′′0 = (ω0,s+1, . . . , ω0n)

and decompose into blocks, the matrices

A :=(A11 A12

A21 A22

), B :=

(B11 B12

B21 B22

), Ω′ =

(Ω′1Ω′2

), Υ =

(Υ1 Υ2

),

where dimA11 = dimB11 = s× s, dim Ω′1 = s×m, dim Υ1 = m× s, we will get asmooth system of the form

p = 2εD[p][α′ −A11p+ εa

(p,−→|εζ|2, v

)− ε2A12

−→|ζ|2]

+ εN/2D1/2[p]a(p,Re ζ, Im ζ, v, ϑ),(6.1)

v = ες[Υ1p+ Γv + εg(p,

−→|εζ|2, v) + ε2Υ2

−→|ζ|2

+ ε(N+1)/2g(p,Re ζ, Im ζ, v, ϑ)],

(6.2)

ζ = D[i(ω′′0 + ε(β′′ + Ω′2v −B21p)) + ε(α′′ −A21p)

]ζ

+ ε2D[− (iB22 +A22)

−→|ζ|2 + h(p,

−→|εζ|2, v)

]ζ + ε(N−2)/2h(p,Re ζ, Im ζ, v, ϑ),

ϑ = ω′0 + ε(β′ + Ω′1v −B11p) + ε2[−B12

−→|ζ|2 + b(p,

−→|εζ|2, v)

]+ εN/2D−1/2[p]b(p,Re ζ, Im ζ, v, ϑ),

whose domain contains the set cl(D1) × ζ ∈ Cn−s : ‖ζ‖ ≤ 1 × Ts, with D1 ⊂Rs+ × Rm being defined by the inequalities

pk >αk

2akk, k ∈ 1, . . . , s, |p| < 2 max1≤k≤s αk

a0+

, ‖v‖ < R.

Let (p(t), v(t), ζ(t), ϑ(t)) be a solution of this system which corresponds to asolution of type (1, . . . , s), so that

(p(t), v(t), ζ(t)) ∈ D1 × ζ ∈ Cn−s : ‖ζ‖ ≤ 1 ∀t > t∗

if t∗ > 0 is large enough.The following is an analogue of Proposition 5.1.

Proposition 6.1. For sufficiently small ε0 > 0 and for any ε ∈ (0, ε0], the system

α′ −A11p+ εa(p, 0, v) = 0, Υ1p+ Γv + εg(p, 0, v)

has a solution

p = p∗(ε) := A−111 α

′ +O(ε), v = v∗(ε) := −Γ−1Υ1A−111 α

′ +O(ε),

such that

p∗k(ε) >2αk3akk

, k ∈ 1, . . . , s, |p∗(ε)| <3 max1≤k≤n αs

2a0+

, ‖v∗(ε)‖ <2R3.

Note that the equilibrium (p∗, v∗) of the system

p = 2εD[p][α′ −A11p+ εa(p, 0, v)],

v = ες[Υ1p+ Γv + εg(p, 0, v)]


is the unique stationary point in D1 of the Morse function

W (p, v) :=s∑i=1

(pi + p∗i ln(

p∗ipi

)− p∗i)

+12〈V ′′(0)(v − v∗), v − v∗〉.

Proposition 6.2. There are positive numbers c7, ς0 and ε0, such that for anyς ∈ (0, ς0), ε ∈ (0, ε0) the following assertion is valid. If (z(t), v(t)) is a solution oftype (1, . . . , s) of system (4.3)–(4.4) and

p(t) :=(|z1(t)|2, . . . , |zs(t)|2

), ζ(t) :=

1ε

(zs+1(t), . . . , zn(t)),

then there exists t∗ > 0, such that√‖p(t)− p∗(ε)‖2 + ‖v(t)− v∗(ε)‖2 <

c7ε(N−2)/2

ς, ‖ζ(t)‖ ≤ c7ε(N−4)/2 ∀t > t∗.

Proof. Since all elements of the matrix A21 are non-positive, then the minimalelement of α′′ − A21p is not less than the minimal element of α′′. Thus, thereexists a constant c8 > 0, such that for all sufficiently small ε > 0 there hold theinequalities

ddt‖ζ(t)‖2 ≥ ‖ζ(t)‖

[ε mins+1≤k≤n

αk‖ζ(t)‖ − c8ε(N−2)/2], ‖ζ(t)‖ ≤ 1 ∀t ≥ t∗,

and consequently,

‖ζ(t)‖ ≤ c8ε(N−4)/2

mins+1≤k≤n αkfor all t ≥ t∗.

Now observe that for

‖ζ‖ ≤ c8ε(N−3)/2

mins+1≤k≤n αk

sub-system (6.1)–(6.2) takes the form

p = 2εD[p] [α′ −A11p+ εa(p, 0, v)] +O(εN/2),

v = ες [Υ1p+ Γv + εg(p, 0, v)] +O(ε(N+3)/2).

Using Lemma 3.2 for the function W (p, v), one can obtain the inequality for p(t)and v(t) precisely in the same way as we did in Proposition 5.2.

If we introduce new variables η ∈ Rs+m and ξ ∈ Rs+m × Cn−s via the formulae

(p, v) = (p∗, v∗) + εση,

then again we come to the perturbation problem for the trivial hyperbolic invarianttorus of the system

η = ε

(−2D[p∗(0)]A11 0

ςΥ1 ςΓ

)η,

ζ = D [i(ω′′0 + ε(β′′ + Ω′2v∗(0)−B21p∗(0))) + ε(α′′ −A12p∗(0))] ζ,

ϑ = ω′0 + ε [β′ + Ω′1v∗(0)−B11p∗(0)] .

The real parts of the eigenvalues of this system’s matrix are non-zero, since theyare defined by the eigenvalues of the matrix

εdiag(−2D[p∗(0)]A11, ςΓ, α′′ −A12p∗(0)).


It is not hard to see that the perturbation terms satisfy the conditions of [19, Lemma2.1]. Using the same arguments as above for the realification of the perturbed sys-tem in the variables η,Re ζ, Im ζ, ϑ, we can prove the existence of an s-dimensionaltruly hyperbolic invariant torus T sε (1, . . . , s), which attracts the trajectories corre-sponding to the solutions of type (1, . . . , s). In the coordinates (p, v, ζ, ϑ) this torusis given by the equations

(p, v) = (p∗(ε), v∗(ε)) + εση∗(ϑ; ε), ζ = ζ∗(ϑ; ε),

where η∗(·; ε) : Ts → Rs+m and ζ∗(·; ε) : Ts → Cn−s are Lipschitzian mappingssatisfying the conditions

maxϑ∈Ts

‖η∗(ϑ; ε)‖ ≤ c8ε(N−2)/2−γ

ς, max

ϑ∈Ts‖ζ∗(ϑ; ε)‖ ≤ c8ε(N−4)/2.

Hence, we have proved the part of statements (3)–(5) from the main theoremconcerning the tori of dimensions less then n. Note, that those forward semi-trajectories in A that are not attracted by stable n-dimensional tori lie on stablemanifolds of truly hyperbolic tori, and thus form the set of zero Lebesgue measure.

7. Excitation of two-frequency oscillations in a system of twocoupled generators due to slow cooling

Let us now provide an example of a hypothetical device where the described phe-nomenon can be observed. Consider a system of two coupled generators (Figure 1).

Figure 1. Coupled generators

Here, the i-th generator consists of the following elements: a thermistor Ri, amagneto resistor ρi, magnetically connected inductors Li, L′i having a negativemutual induction Mi = γi

√LiL′i, a capacitor Ci, and an active feedback element

Ai. We suppose that the I-V characteristics χi of Ai is a smooth function of voltagedifference Vi on L′i and admits the expansion

χi = χi(Vi) := χi1Vi − χi3V 3i +O(V 4

i ), χi1 > 0, χi3 > 0.

We assume that the i-th thermistor has a positive temperature coefficient and thatits resistance depends on the thermistor’s temperature Ti by the relation

Ri = Ri(Ti) := εRi0 + µRi1Ti, i ∈ 1, 2,


where Ri0, Ri1 are positive constants and ε and µ are small parameters. We alsosuppose the inductances Li, L′i and the capacities Ci to be constant.

The resistance of the magneto resistor depends on an external magnetic field.If the latter is orthogonal to the direction of the current in the resistor, then thechange of its resistance is approximately proportional to the square of the magneticfield magnitude [25].

The generators interact with each other in the following way. The current Ii inthe inductor Li produces a magnetic field of a magnitude proportional to Ii, andthis magnetic field influences the resistance of the magneto resistor ρj . Thus, it isnatural to consider the case where

ρj = ρj(Ii) = ρj0 + ρj2(LiIi)2, ρj0 > 0, ρj2 > 0, i ∈ 1, 2. (7.1)

Taking into account the Newton law of cooling and the Joule – Lenz law of ohmicheating and assuming the environment temperature to be zero, we adopt the nextequation for the resistor Ri temperature change

Ti = −µkiTi +KiI2i Ri, (7.2)

where ki and Ki are some positive coefficients.Our goal is to show that under an appropriate choice of the generators param-

eters one can observe such a phenomenon. If the described device of the coupledgenerators with the sufficiently high initial temperatures Ti(0) of the resistancesRi is placed into the environment, then first for a long period of time this deviceremains in the sleep mode. However, when the resistances temperatures drop al-most to zero, the device wakes up and after a transient process, in general, it startsproducing two-frequency oscillations. Such a kind of behavior can be treated as aphenomenon of dynamical bifurcation.

Denote by qi the charge of the capacitor Ci, and let Ii, ICi , Iρi and IAi stand forthe currents through the resistor Ri, the capacitor Ci, the magneto resistor ρi andthe active element Ai respectively. The Kirchhoff laws yield that

Ii + ICi + Iρi = IAi = χi, (7.3)

LiIi +RiIi =qiCi

= ρiIρi . (7.4)

Since qi = ICi , by differentiation of (7.4) we obtain

LiIRi +RiIi + RiIi =ICiCi

.

But from (7.4) and (7.3) we can find

Iρi = [LiIi +RiIi]/ρi,

ICi = χi − Ii −1ρi

[LiIi +RiIi].

Hence,

Ii +RiLiIi +

IiLiCi

+1

ρiLiCi[LiIi +RiIi] =

χiLiCi

− RiLiIi,

and on account of (7.2)

Ii + [RiLi

+1

ρiCi]Ii +

1LiCi

[1 +Riρi

]Ii =χiLiCi

− µRi1Li

[−µkiTi +KiI2i Ri]Ii.


Introducing new variables Ii = √ρj0wi, Ti = Ri0ui/(ςRi1), imposing the constraintςε = µ with the constant ς playing the same role as in section 4 and taking intoaccount (7.1) and (7.2), we finally reach the system

wi +[εRi0(1 + ui)

Li+

1Ciρi0(1 + ρi2L2

jw2j )

]wi +

1LiCi

[1 +

εRi0(1 + ui)ρi0(1 + ρi2L2

jw2j )

]wi

=1

LiCi√ρj0

χi(Mi√ρj0wi)−

µεRi0Li

[−kiui +KiRi1(1 + ui)ρj0w2i ]wi,

ui = µ[−kiui +Ri1Kiρj0(1 + ui)w2i ], i ∈ 1, 2, j ∈ 1, 2, i 6= j.

This system can be represented in the form (2.3) if the parameters of the generatorsare chosen in such a way, that

χi1Mi

LiCi− 1Ciρi0

= εbi, i ∈ 1, 2,

where b1, b2 are positive constants satisfying the inequalities

bi > Ri0/Li, i ∈ 1, 2. (7.5)

In fact, we have

ω0i(u) =√

1LiCi

+O(ε), λi(u) = bi −Ri0(1 + ui)

Li, i ∈ 1, 2,

and

Fi(w1, w2, w1, w2, u, 0)

=∑

k+l+m+n=3

fi,jklmwj1w

k1w

l2w

m2 +O(‖w‖4 + ‖w‖4), i ∈ 1, 2,

where the only non-zero coefficients are

f1,0300 = −χ13ρ20M31

LiCi, f1,0120 =

ρ12L21

C1ρ10,

f2,2001 =ρ22L

22

C2ρ20, f2,0003 = −χ23ρ10M

32

LiCi.

Therefore, performing the change of variables (2.2), we obtain

F (x, u, ςε) =

0

1ω01(u)F1(x1, x3, ω01(u)x2, ω02(u)x4, u, 0)

01

ω02(u)F2(x1, x3, ω01(u)x2, ω02(u)x4, u, 0)

+O(ε).

It is easily seen the eigenvectors of the matrix J(u) are

s+1 =

−i100

, s+2 =

00−i1

, s−1 =

i100

, s−2 =

00i1

.

If we now introduce new complex variables z1, z2 ∈ C via

x1 = −iz1 + iz1, x2 = z1 + z1, x3 = −iz2 + iz2, x4 = z2 + z2,


we will be able to find the elements of the matrix A(u) by extraction of the resonantterms from cubic nonlinearities:

12ω01(u)

F1(−iz1 + iz1,−iz2 + iz2, ω01(u)(z1 + z1), ω02(u)(z2 + z2), u, 0)

= −a11(u)|z1|2z1 − a12(u)|z2|2z1 + [nonresonant terms]1

2ω02(u)F2(−iz1 + iz1,−iz2 + iz2, ω01(u)(z1 + z1), ω02(u)(z2 + z2), u, 0)

= −a21(u)|z1|2z2 − a22(u)|z2|2z2 + [nonresonant terms].

It turns out that

a11(u) = −12f1,2100 −

32ω2

01(u)f1,0300,

a12(u) = −f1,0120 − ω202(u)f1,0102,

a21(u) = −f2,2001 − ω201(u)f2,0201,

a22(u) = −12f2,0021 −

32ω2

02(u)f2,0003.

In our case, when ε = 0, these elements does not depend on u:

a11 =3χ13ρ20M

31

2(C1L1)2, a12 = −ρ12L

22

C1ρ10,

a21 = −ρ22L21

C2ρ20, a22 =

3χ23ρ10M32

2(C2L2)2.

Hence, when ε = 0, the positive definiteness condition of the symmetric part ofthe matrix A takes the form

9χ13χ23(ρ10ρ20M1M2)3 >[L1L2(ρ12ρ20C2L

22 + ρ10ρ22C1L

21)]2. (7.6)

Since g(u) = (−k1u1,−k2u2), the Morse function can be chosen as V (u) =u2

1 + u22. It has a unique stationary point u∗ = (0, 0). The instability and stability

sets are defined as follows

V+ = (u1, u2) ∈ R2 : u21 + u2

2 < V ∗, ui < biLi0/Ri0 − 1, i ∈ 1, 2,V− = (u1, u2) ∈ R2 : u2

1 + u22 < V ∗, ui > biLi0/Ri0 − 1, i ∈ 1, 2,

where V ∗ > 0 is large enough.Thus, if the numbers

√C1L1,

√C2L2 are rationally independent and condi-

tions (7.5) and (7.6) fulfill, then hypotheses (H1)–(H5) hold, and Theorem 2.2implies that under the appropriate choice of ς0 > 0, 0 < ς∗ < ς0, ε0 > 0 the afore-mentioned changes in behavior of the coupled generators can actually be observed,once ε ∈ (0, ε0), µ ∈ (ς∗ε, ς0ε).

8. Addendum

Consider the formal system

x = J(u)x+∑i≥0

µiFi(x, u),

u = µ[g(u) +

∑i≥0

µiGi(x, u)] (8.1)


obtained from (2.3) by expanding its right-hand sides into the Taylor series expan-sions in powers of µ. Our goal is to simplify this system with aid of the formalchange of variables

x = y +∑i≥0

µiXi(y, v), u = v + µ∑i≥0

µiUi(y, v). (8.2)

Construction of such a transformation consists in solving homological equationsof the form

LJ(v)y[Yj(v)yj ] = Pj(v)yj −Rj(v)yj , (8.3)

∂J(v)y[Zj(v)yj ] = Qj(v)yj − Sj(v)yj . (8.4)

Here Pj(v)yj and Qj(v)yj are known j-th order homogeneous forms which take val-ues in R2n and Rm respectively and smoothly depend on the parameter v, whereas∂J(v)y and LJ(v)y are, respectively, the directional and the Lie derivatives along thevector field J(v)y. Namely,

∂J(v)yZ(y) :=∂Z(y)∂y

J(v)y ∀Z(·) ∈ C1(R2n; Rm),

LJ(v)yY (y) :=∂Y (y)∂y

J(v)y − J(v)Y (y) ∀Y (·) ∈ C1(R2n; R2n).

The forms Yj(v),Rj(v)yj , Zj(v) and Sj(v)yj are determined in such a way, thatthey smoothly depend on v and satisfy the corresponding equations.

If j = 0, equation (8.3) has obvious solutions

R0(v) ≡ 0, Y0(v) = −J−1(v)P0(v), S0(v) = Q0(v), Z0(v) = 0. (8.5)

In the case when j ≥ 1, in the same manner as in [37], we can introduce a suitablebasis in the space of vector-valued polynomial forms. Note, that the matrix J(v)has constant linearly independent eigenvectors s±j ∈ C2n, j = 1, . . . , n, such that

J(v)s±j = [ελj(v)± iωj(v)]s±j ,

with vectors s−j and s+j being complex conjugate for all j = 1, . . . , n. Denote by

S the matrix with the columns s+1 , . . . , s

+n , s−1 , . . . , s

−n , and define the homogeneous

formssq(y) := [S−1y]q, e±j,q(y) = sq(y)s±j ,

where q := (q1, . . . , q2n) ∈ Z2n+ and xq := xq11 · · ·x

q2n2n for x = (x1, . . . , x2n). This

gives us the following expansions.

Pj(v)yj =n∑k=1

∑|q|=j

[P+k,q(v)e+

k,q(y) + P−k,q(v)e−k,q(y)], Qj(v)yj =∑|q|=j

sq(y)Qq(v),

Rj(v)yj =n∑k=1

∑|q|=j

[R+k,q(v)e+

k,q(y) +R−k,q(v)e−k,q(y)], Sj(v)yj =∑|q|=j

sq(y)Sq(v).

Proposition 8.1. Let DN ⊂ Rm be such a domain, that for any k ∈ 1, . . . , nand σ ∈ 0, 1 the equality

minv∈cl(DN )

∣∣ n∑l=1

(ql − ql+n − σδkl)ω0l(v)∣∣ = 0,


where q ∈ Z2n+ , 1 ≤ |q| :=

∑2nk=1 qk ≤ N and δkl is Kronecker’s delta, holds if and

only ifql − ql+n − σδkl = 0 ∀l ∈ 1, . . . , n.

Suppose also that the forms Pj(v)yj and Qj(v)yj smoothly depend on v ∈ cl(DN )and assign

R±k,q(v) =

P±k,q(v) if

∑nl=1 |ql − ql+n ∓ δkl| = 0,

0 if∑nl=1 |ql − ql+n ∓ δkl| 6= 0,

Sq(v) =

Qq(v) if

∑nl=1 |ql − ql+n| = 0,

0 if∑nl=1 |ql − ql+n| 6= 0.

Then, for sufficiently small ε0 > 0 and for all ε ∈ [0, ε0], there exist an R2n-valuedform Yj(v)yj and an Rm-valued form Zj(v)yj, which satisfy equations (8.3) and(8.4) respectively. The coefficients of these forms are smooth functions in cl(DN ).

Proof. Since

S−1J(v)S = diag[ελ1(v) + iω1(v), . . . , ελn(v) + iωn(v), ελ1(v)

− iω1(v), . . . , ελn(v)− iωn(v)],

we have

∂J(v)ysq(y) =ddt

∣∣t=0

[(S−1eJ(v)ty)q]

=[ε

n∑l=1

(ql + ql+n)λl(v) + in∑l=1

(ql − ql+n)ωl(v)]sq(y)

and

LJ(v)ye±k,q(y) =

ddt

∣∣t=0

e−J(v)te±k,q(eJ(v)ty

)=[ε

n∑l=1

(ql + ql+n − δkl)λl(v) + in∑l=1

(ql − ql+n ∓ δkl)ωl(v)]e±k,q(y).

After expanding the forms

Yj(v)yj =n∑k=1

∑|q|=j

[Y +k,q(v)e+

k,q(y) + Y −k,q(v)e−k,q(y)],

Zj(v)yj =∑|q|=j

sq(y)Zq(v),

the homological equations are reduced to[ε

n∑l=1

(ql + ql+n − δkl)λl(v) + in∑l=1

(ql − ql+n ∓ δkl)ωl(v)]Y ±k,q(v)

= P±k,q(v)−R±k,q(v),[ε

n∑l=1

(ql + ql+n)λl(v) + in∑l=1

(ql − ql+n)ωl(v)]Zq(v) = Qq(v)− Sq(v).

Taking into account the definitions of R±k,q(v) and Sq(v), these equations are solublefor any ε ∈ [0, ε0] with sufficiently small ε0 > 0, and, as a consequence, the same is


true for equations (8.3). Although the coefficients Y ±k,q(v) and Zq(v) are complex-valued, one can make the forms Yj(v)yj and Zj(v)yj take values, respectively, inR2n and Rm by setting Y ±k,q(v) ≡ 0 for any q, such that

∑nl=1 |ql − ql+n ∓ δkl| = 0,

and Zq(v) ≡ 0 for any q, such that∑nl=1 |ql−ql+n| = 0. The smoothness properties

of Yj(v)yj ,Zj(v)yj are obvious.

Remark 8.2. For q ∈ Z2n+ with |q| = 1, the equality

∑nl=1 |ql − ql+n ∓ δkl| = 0 is

satisfied if and only if q = e±k , where e+k ∈ Z2n

+ (e−k ∈ Z2n+ ) is a vector whose k-th

((k + n)-th) coordinate equals 1 while the other are 0. Hence, in such case,

R±k,q(v) 6= 0 if and only if q = e±k .

Let

y = J(v)y +∑i≥0

µiHi(y, v),

v = µ[g(v) +

∑i≥0

µiCi(y, v)],

be the system obtained from (8.1) by means of the formal change of variables (8.2).In view of (1.3) and the definition of g(v), we have

F0(y, v) = O(y2), G0(y, v) = O(y).

Therefore, we require that

X0(y, v) = O(y2), H0(y, v) = O(y2), C0(y, v) = O(y), U0(y, v) = O(y).

Substituting (8.2) in (8.1) and equating coefficients near like powers of µ, we obtainthe following chain of homological equations for the unknown coefficients

LJ(v)yX0(y, v) = F0(y +X0(y, v), v)− ∂X0(y, v)∂y

H0(y, v)−H0(y, v),

∂J(v)yU0(y, v) = G0(y +X0(y, v), v)− ∂U0(y, v)∂y

H0(y, v)− C0(y, v),

and, for i > 0,

LJ(v)yXi(y, v)

=∂Xi−k(y, v)

∂yHk(y, v)− ∂Xi−1(y, v)

∂vg(v)

−i−1∑k=0

∂Xi−k−1(y, v)∂v

Ck(y, v) +[∂F0(x, v)

∂xXi(y, v)

]x=y+X0(y,v)

+1i!∂i

∂µi∣∣µ=0

[J(v + µ

i−1∑j=0

µjUj(y, v))(y +

i−1∑k=0

µkXk(y, v))]

+1i!∂i

∂µi∣∣µ=0

i∑l=0

µlFl

(y +

i−1∑k=0

µkXk(y, v), v + µ

i−1∑j=0

µjUj(y, v))−Hi(y, v),

∂J(v)yUi(y, v)

= −i∑

k=0

∂Ui−k(y, v)∂y

Hk(y, v)− ∂Ui−1(y, v)∂v

g(v)


−i−1∑k=0

∂Ui−k−1(y, v)∂v

Ck(y, v) +[∂G0(v, x)

∂xXi(y, v)

]x=y+X0(y,v)

+1i!∂i

∂µi∣∣µ=0

g(v + µ

i−1∑j=0

µjUj(y, v))

+1i!∂i

∂µi∣∣µ=0

i∑l=0

µlGl

(y +

i−1∑k=0

µkXk(y, v), v + µ

i−1∑j=0

µjUj(y, v))− Ci(y, v).

On account of (1.3) and the definition of g(v), we have the following expansions.

F0(y, v) =∑j≥2

F0j(v)yj , Fi(y, v) =∑j≥0

Fij(v)yj , i ≥ 1,

G0(y, v) =∑j≥1

G0j(v)yj , Gi(y, v) =∑j≥0

Gij(v)yj , i ≥ 1.

Thus, the unknown functions can be sought in the form

X0(y, v) =∑j≥2

X0j(v)yj , Xi(y, v) =∑j≥0

Xij(v)yj , i ≥ 1,

H0(y, v) =∑j≥2

H0j(v)yj , Hi(y, v) =∑j≥0

Hij(v)yj , i ≥ 1,

U0(y, v) =∑j≥1

U0j(v)yj , Ui(y, v) =∑j≥0

Uij(v)yj , i ≥ 1,

C0(y, v) =∑j≥1

C0j(v)yj , Ci(y, v) =∑j≥0

Cij(v)yj , i ≥ 1,

It is easily seen that the coefficients of these expansions satisfy the chain of equations

LJ(v)y[Xij(v)yj ] = Pij(v)yj −Hij(v)yj , ∂J(v)y[Uij(v)yj ] = Qij(v)yj − Cij(v)yj ,

where Pij(v) and Qij(v) can be determined subsequently. In fact, P02(v) := F02(v),Q01(v) := G01(v), and now, if v ∈ DN , one can use Proposition 8.1 to get H02 = 0,X02, C01 = 0, U01, and then subsequently find P0j , H0j , X0j for j = 3, . . . , N ,and Q0j , C0j , U0j for j = 2, . . . , N . If 0 ≤ k < i, l ≤ N and the coefficientsXkl(v), Hkl(v), Ukl(v), Ckl(v) are already known, then one can determine Pi0(v)and subsequently find Hij(v), Xij(v), Pi,j+1(v) for j = 0, . . . , N . (Note that (8.5)yields Hi0(v) = 0.) After that, Qi0(v) can be determined, and subsequently Cij(v),Uij(v), Qi,j+1(v) may be found for j = 0, . . . , N . At last, for any j > N , we assignHij(v) ≡ Pij(v), Xij(v) ≡ 0, Cij(v) ≡ Qij(v) and Uij(v) ≡ 0.

This result can be summed up as the following proposition.

Proposition 8.3. Suppose that P ≥ 2, N ≥ 3 and DN satisfy the conditions ofProposition 8.1. Then there exist δ0 > 0 and µ0 > 0, such that for any ε ∈ [0, ε0]the smooth diffeomorphic change of variables

x = y +N∑j=2

X0j(v)yj +P∑i=1

µiN∑j=0

Xij(v)yj ,

u = v +N∑j=1

U0j(v)yj +P∑i=1

µiN∑j=0

Uij(v)yj


defined on the set (y, v, µ) : ‖y‖ < δ0, v ∈ DN , µ ∈ [0, µ0] transforms system (2.3)into

y = J(v)y +N∑j=3

H0j(v)yj +P∑i=1

µiN∑j=1

Hij(v)yj +O(‖y‖N+1 + µP+1),

v = µ[g(v) +

N∑j=2

C0j(v)yj +P∑i=1

µiN∑j=0

Cij(v)yj +O(‖y‖N+1 + µP+1)].

(8.6)

Here the homogeneous forms in the right-hand sides admit the expansions

Hij(v)yj =n∑k=1

[∑|q|=j

+h+i,k,q(v)e+

k,q(y) +∑|q|=j

−h−i,k,q(v)e−k,q(y)

],

Cij(v)yj =n∑k=1

∑|q|=j

′sq(y)ci,q(v),

where h±i,k,q(·) ∈ C∞(DN ; C), ci,q(·) ∈ C∞(DN ; Cm) and the summations∑+|q|=j,∑−

|q|=j,∑′|q|=j are performed on all vectors q ∈ Z2n

+ with |q| = j whose componentssatisfy, respectively, the equalities ql = ql+n + 1, ql+n = ql + 1 and ql = ql+n for alll ∈ 1, . . . , n.

Remark 8.4. In view of Remark 2.1, under hypothesis (H2) the assertion of Propo-sition 8.3 is true for N = 3 and D3 = V. If hypothesis (H3) is valid, then thisproposition is correct in a sufficiently small neighborhood of any stationary pointv∗ ∈ W.

On account of Remark 8.2,

sq(S(z, z)) = (z, z)q, S−1e+k,q(Sz) = (z, z)qe+

k , S−1e+k,q(Sz) = (z, z)qe−k ,

S−1Hi1(v)(S(z, z))1 =n∑k=1

[zkh

+

i,k,e+k

(v)e+k + zkh

−i,k,e−k

(v)e−k],

which means that the matrices of each of the linear form S−1Hi1(v)(S(z, z))1 arediagonal and the change of variables

y = S(z, z) =n∑k=1

(zksk + zksk), ‖z‖ < ‖S‖δ0 =: δ1,

reduces system (8.6) to the form (2.4), where

ηj,k(v) := h+

i,k,e+k

(v), hj,k,p(v) := h+

j,k,(p,p)+e+k

(v), gj,p(v) := cj,(p,p)(v),

and (p,p) := (p1, . . . , pn, p1, . . . , pn).

9. Summary

In this article we have examined a system of oscillators with weak and slowcoupling which demonstrates a dynamic bifurcation of multi-frequency oscillations.Having adopted results of the static bifurcation theory, we have shown that whenthe system’s parameters slowly evolve and the static parameters are sufficientlysmall, then certain general conditions guarantee occurrence of the following tran-sient process for typical forward trajectories within a small neighborhood of theslow surface. While the slow component u(t) is far from the stationary points of


the Morse function V and lies inside the stability zone of the fast subsystem, thefast component (w(t), w(t)) exhibits damping oscillations with the amplitude ap-proaching zero. However, as soon as u(t) leaves the stability zone and later entersthe zone of instability of the fast subsystem, the amplitude starts to grow andeventually the forward trajectory is attracted by an invariant torus, which meansestablishment of some multi-frequency oscillatory regime.

It was shown that almost all forward trajectories, in terms of the Lebesguemeasure, starting from the neighborhood of the slow surface demonstrate sucha behavior. More than that, they are attracted by trajectories on the stable n-dimensional invariant tori, whereas all other forward trajectories of the system lieon the stable manifolds of hyperbolic tori of dimensions less than n. This enablesus to easily categorize the trajectories by the type of their ultimate behavior.

At last, we have also considered a practical example which depicts occurrence ofthe multi-frequency bifurcation in a circuit of two coupled oscillators.

References

[1] V. S. Afrajmovich, V. I. Arnold, Yu. S. Il’yashenko, L. P. Shil’nikov; Dynamical Systems V.Encyclopaedia of Mathematical Sciences. Springer-Verlag, 1994.

[2] D. V. Anosov; On the development of the theory of dynamical systems during the past quar-

ter century. Surveys in modern mathematics, London Math. Soc. Lecture Note Ser., 321,Cambridge Univ. Press, Cambridge, 2005, 70–185.

[3] O. D. Anosova; Invariant manifolds and dynamic bifurcations. Russ. Math. Surv. 60 (1)

(2005) 151–153.[4] P. Ashwin, S. Coombes, R. Nicks; Mathematical frameworks for oscillatory network dynamics

in neuroscience. J. Math. Neurosci. 6:2 (2016).

[5] E Benoıt (ed.); Dynamic bifurcations. Lecture Notes in Mathematics. 1493. Springer-Verlag,

Berlin, 1991.[6] Yu. N. Bibikov; Hopf bifurcation for quasiperiodic motion. Differ. Equ., 16 (1981) 977–982.

[7] Yu. N. Bibikov; Bifurcation of a stable invariant torus from an equilibrium. Math. Notes 48

(1) (1990), 632–635.[8] Yu. N. Bibikov; Multi-frequency non-linear oscillations and their bifurcations (in Russian).

Leningrad Gos. Univ., Leningrad, 1991.

[9] N. N. Bogolyubov, N. M. Krylov; Application of the methods of non-linear mechanics to thetheory of stationary oscillations (in Russian). Kyiv: Publ. Acad. of Sci. Ukr. SSR, 1934.

[10] N. N. Bogolyubov, Yu. A. Mitropol’skij, A. M. Samojlenko; Methods of accelerated con-

vergence in nonlinear mechanics. Delhi, India: Hindustan Publishing Corporation; Berlin-Heidelberg-New York: Springer-Verlag, 1976.

[11] V. F. Butuzov, N. N. Nefedov, K. R. Schneider; Singularly perturbed problems in case of

exchange of stabilities. J. Math. Sci., 121 (1) (2004), 1973–2079.[12] P. Cartier; Singular perturbations of ordinary differential equations and non-standard anal-

ysis (in Russian). Uspekhi Mat. Nauk., 39 (2) (1984) 57–76.[13] Yu. P. Emelianova, A. P. Kuznetsov, I. R. Sataev, L. V. Turukina; Synchronization and

multi-frequency oscillations in the low-dimensional chain of the self-oscillators. Physica D:Nonlinear Phenomena, 244 (1) (2013), 36–49.

[14] T. Endo, K. Kamiyama, M. Komuro; Bifurcation of quasi-periodic oscillations in mutuallycoupled hard-type oscillators: demonstration of unstable quasi-periodic orbits. Internat. J.

Bifur. Chaos Appl. Sci. Engrg., 22 (6) (2012).[15] N. Fenichel; Persistence and smoothness of invariant manifolds and flows. Indiana Univ.

Math., 21 (3) (1971), 193–226.[16] F. R. Gantmacher; The Theory of Matrices. American Mathematical Society, 1998.[17] D. E. Gilsinn; Constructing invariant tori for two weakly coupled van der Pol oscillators

Nonlinear Dynamics, 4 (3) (1993), 289–308.

[18] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov; Blue sky catastrophe in relaxation systems withone fast and two slow variables. Differ. Equ., 44 (2) (2008), 161–175.


[19] J. K. Hale; Integral manifolds of perturbed differential systems. Ann. of Math. Second Series,

73 (3) (1961), 496–531.

[20] H. Hanssmann; A Survey on Bifurcations of Invariant Tori. New Advances in CelestialMechanics and Hamiltonian Systems. HAMSYS-2001 (2004), 109–121.

[21] P. Hartman; Ordinary differential equations. John Wiley and Sons, New York-London-

Sydney, 1964.[22] F. C. Hoppensteadt, E. M. Izhikevich; Weakly connected neural networks. Applied Mathe-

matical Sciences, 126. New York, NY: Springer. xvi, 1997.

[23] F. C. Hoppensteadt, E. M. Izhikevich; Slowly coupled oscillators: phase dynamics and syn-chronization. SIAM J. Appl. Math., 63 (6) (2003), 1935–1953.

[24] G. Iooss, D. D. Joseph; Elementary Stability and Bifurcation Theory. Springer-Verlag New

York, 1990.[25] P. S. Kireev; Semiconductor physics, 2nd ed. Mir Publishers, Moscow, 1978.

[26] A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov; On one bifurcation in relaxation systems.Ukranian Math. J., 60 (1) (2008), 66–77.

[27] Yu. A. Kuznetsov; Elements of applied bifurcation theory. Springer, New York, 1998.

[28] S. Liebscher; Dynamics near manifolds of equilibria of codimension one and bifurcationwithout parameters. Electron. J. Differential Equations, 2011 (63) (2011), 1–12.

[29] J. Marsden, J. Scheurle; Bifurcation to quasi-periodic tori in the interaction of steady state

and hopf bifurcations. SIAM J. Math. Anal., 15 (6) (1984), 1055–1074.[30] D. T. Mook, A. H. Nayfeh; Nonlinear oscillations. John Wiley & Sons, New York, 1979.

[31] Yu. A. Mitropol’skij, A. M. Samojlenko; Multifrequency oscillations of weakly nonlinear

second-order systems. Ukrainian Math. J., 28 (1977), 572–586.[32] Yu. I. Neimark; On some cases of dependency of periodic motions on parameters (in Rus-

sian). Dokl. AN SSSR., 129 (4) (1959), 736–739.

[33] A. I. Nejshtadt; Asymptotic investigation of the loss of stability by an equilibrium as a pairof eigenvalues slowly cross the imaginary axis. Uspkhi Mat. Nauk, 40 (5) (1985), 190–191

[34] A. I. Nejshtadt; Persistence of stability loss for dynamical bifurcations. I. Differ. Equ., 23(12) (1987), 1385–1391.

[35] A. I. Nejshtadt; On deferring the loss of stability for dynamic bifurcations. II. Differ. Equ.,

24 (2) (1988), 171–176.[36] H. Nikaido; Convex Structures and Economic Theory. Academic Press, New York and Lon-

don, 1968.

[37] I. O. Parasyuk, B. V. Repeta, A. M. Samoilenko; Dynamical bifurcation of multifrequencyoscillations in a fast-slow system. Ukrainian Math. J., 67 (7) (2015), 1008–1037.

[38] I. O. Parasyuk, B. V. Repeta; Hyperbolic invariant tori of a fast-slow system exhibiting the

dynamical bifurcation of multi-frequency oscillations (in Ukrainian). Nelinijni Kolyvannya,19 (1) (2016), 101–121.

[39] R. Petryshyn, A. Samoilenko; Multifrequency oscillations of nonlinear systems. Mathematics

and its Applications. Kluwer Academic Publishers, Dordrecht, 2004.[40] D. Rachinskii, K. Schneider; Dynamic Hopf bifurcations generated by nonlinear terms. J.

Differential Equations, 210 (1) (2005), 65–86.[41] R. Sacker; On invariant surfaces and bifurcation of periodic solutions of ordinary differential

equations. Report IMM-NYU 333, New York University, 1964.

[42] R. Sacker; A new approach to perturbation theory of invariant surfaces. Comm. Pure Appl.Math., 18 (1965) 717–732.

[43] A. M. Samoilenko; Elements of mathematical theory of multi-frequency oscillations. KluwerAcademic Publishers, Dordrecht, 1991.

[44] A. M. Samoilenko; Perturbation theory of smooth invariant tori of dynamical systems. Non-

linear Anal., 30 (5) (1997), 3121–3133.

[45] S. A. Shchegolev; On some classes of solutions of two-dimensional multifrequency quasilineardifferential systems. Differ. Equ., 40 (10) (2004), 1413–1418.

[46] A. Shilnikov, L. Shilnikov, D. Turaev; Blue-sky catastrophe in singularly perturbed systems.Mosc. Math. J., 5 (1) (2005), 269–282.

[47] M. A. Shishkova; Examination of a system of differential equations with a small parameter

in the highest derivatives. Sov. Math., Dokl., 14 (1973), 483–487.


Igor Parasyuk

Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv,

64/13, Volodymyrska Street, City of Kyiv, 01601, UkraineE-mail address: [email protected]

Bogdan RepetaFaculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv,

64/13, Volodymyrska Street, City of Kyiv, 01601, Ukraine

E-mail address: [email protected]

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